Doping-induced in-plane anisotropy of bond-stretching phonon softening in oxychloride Ca$_{2-x}$CuO$_{2}$Cl$_{2}$ compounds

We measure the dispersion of the Cu-O bond-stretching phonon mode in the high-temperature superconducting parent compound Ca2CuO2Cl2. Our density functional theory calculations predict a cosine-shaped bending of the dispersion along both the (ξ00) and (ξξ0) directions, while comparison with previous results on Ca1.84CuO2Cl2 show it only along (ξ00), suggesting an anisotropic effect which is not reproduced in calculation. Comparison with isostructural La2−xSrxCuO4 and literature suggests that these calculations reproduce well the dispersion in the overdoped regime.